English
In a Dedekind domain, the localization at every nonzero prime is a discrete valuation ring.
Русский
В Дедекендовой домене локализация по любому ненулевомуprime является дискретной ценной петлей.
LaTeX
$$$\text{If } A \text{ is Dedekind, then } \mathrm{IsDiscreteValuationRing}(\mathrm{Localization.AtPrime } P)$$$
Lean4
/-- In a Dedekind domain, the localization at every nonzero prime ideal is a DVR. -/
theorem isDiscreteValuationRing_of_dedekind_domain [IsDedekindDomain A] {P : Ideal A} (hP : P ≠ ⊥) [pP : P.IsPrime]
(Aₘ : Type*) [CommRing Aₘ] [IsDomain Aₘ] [Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P] :
IsDiscreteValuationRing Aₘ := by
classical
letI : IsNoetherianRing Aₘ := IsLocalization.isNoetherianRing P.primeCompl _ IsDedekindRing.toIsNoetherian
letI : IsLocalRing Aₘ := IsLocalization.AtPrime.isLocalRing Aₘ P
have hnf := IsLocalization.AtPrime.not_isField A hP Aₘ
exact ((IsDiscreteValuationRing.TFAE Aₘ hnf).out 0 2).mpr (IsLocalization.AtPrime.isDedekindDomain A P _)
